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## G = C5×Dic5order 100 = 22·52

### Direct product of C5 and Dic5

Aliases: C5×Dic5, C52C20, C10.C10, C525C4, C10.4D5, C2.(C5×D5), (C5×C10).1C2, SmallGroup(100,6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C5×Dic5
 Chief series C1 — C5 — C10 — C5×C10 — C5×Dic5
 Lower central C5 — C5×Dic5
 Upper central C1 — C10

Generators and relations for C5×Dic5
G = < a,b,c | a5=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >

Permutation representations of C5×Dic5
On 20 points - transitive group 20T25
Generators in S20
(1 5 9 3 7)(2 6 10 4 8)(11 17 13 19 15)(12 18 14 20 16)
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)
(1 16 6 11)(2 15 7 20)(3 14 8 19)(4 13 9 18)(5 12 10 17)

G:=sub<Sym(20)| (1,5,9,3,7)(2,6,10,4,8)(11,17,13,19,15)(12,18,14,20,16), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,16,6,11)(2,15,7,20)(3,14,8,19)(4,13,9,18)(5,12,10,17)>;

G:=Group( (1,5,9,3,7)(2,6,10,4,8)(11,17,13,19,15)(12,18,14,20,16), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,16,6,11)(2,15,7,20)(3,14,8,19)(4,13,9,18)(5,12,10,17) );

G=PermutationGroup([[(1,5,9,3,7),(2,6,10,4,8),(11,17,13,19,15),(12,18,14,20,16)], [(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20)], [(1,16,6,11),(2,15,7,20),(3,14,8,19),(4,13,9,18),(5,12,10,17)]])

G:=TransitiveGroup(20,25);

C5×Dic5 is a maximal subgroup of
C523C8  Dic52D5  C5⋊D20  C522Q8  D5×C20  C522Dic3  He55C4  C50.C10  He56C4
C5×Dic5 is a maximal quotient of
He55C4  C50.C10

40 conjugacy classes

 class 1 2 4A 4B 5A 5B 5C 5D 5E ··· 5N 10A 10B 10C 10D 10E ··· 10N 20A ··· 20H order 1 2 4 4 5 5 5 5 5 ··· 5 10 10 10 10 10 ··· 10 20 ··· 20 size 1 1 5 5 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 5 ··· 5

40 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + - image C1 C2 C4 C5 C10 C20 D5 Dic5 C5×D5 C5×Dic5 kernel C5×Dic5 C5×C10 C52 Dic5 C10 C5 C10 C5 C2 C1 # reps 1 1 2 4 4 8 2 2 8 8

Matrix representation of C5×Dic5 in GL2(𝔽11) generated by

 9 0 0 9
,
 8 0 0 7
,
 0 10 1 0
G:=sub<GL(2,GF(11))| [9,0,0,9],[8,0,0,7],[0,1,10,0] >;

C5×Dic5 in GAP, Magma, Sage, TeX

C_5\times {\rm Dic}_5
% in TeX

G:=Group("C5xDic5");
// GroupNames label

G:=SmallGroup(100,6);
// by ID

G=gap.SmallGroup(100,6);
# by ID

G:=PCGroup([4,-2,-5,-2,-5,40,1283]);
// Polycyclic

G:=Group<a,b,c|a^5=b^10=1,c^2=b^5,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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